December 08, 2017 Friday
Bedtime Story
Initial Values by Backtracking
Of course, at this point you will surely
raise the question – “Pan narrans, what about the initial value?”
How does the machine get the initial value
that would start the whole process?
Surely the first solution cannot be arrived
at without actually solving the equation manually.
Well, there seems to be a way out of it.
By examining the polynomial equation in its
totality once can get a very decent knowledge of the initial values.
The coefficients of the polynomial will
provide the initial values by a method known as backtracking.
Backtracking these days is a very well
general algorithm for finding solutions to some computational problems.
In our case of difference engine, the
initial value of the first column will be assigned the value of the very first
coefficient of the polynomial function.
Consider for this instance this generalized
polynomial equation of degree n.
f(x) = anxn + an-1xn-1
+ … + a2x2 + a1x + a0
So in this n degree polynomial there are
constant coefficients a0, a1, a2,…, an
Now from these constant coefficients the
initial values can be assigned using the method of backtracking.
The first column will be assigned the value
a0 which will help in the computation of f(0).
The second column will then start
backtracking and get its value by the operation f(1) – f(0).
It is true that this method is not highly
accurate and moreover, the small errors generated by this approximation would
gradually accumulate.
This cumulating error will eventually lead
the series to diverge from the true values of the function.
There, of course, also exists a cure for
this malady but that and its remedy is not the contention of our storyline.
I am after all not an electronic engineer
whose job is to fix the problem of diverging series.
I am Pan narrans, the bed time story teller
whose goal is make each night or may be each day in case of Western
hemispherians (this is an originally coined word) you get not only to learn something unique but
to get good sleep as well that often accompanies mathematical stories.
Now that we understand the mathematics that
lies at the heart of the difference engine, which also admirably explains the
name of this device, let us have a look at the mechanics of this machine.
Now you very well understand that it is one
thing to conceptualize in mind the mathematics of such an intricate device, but
it is a totally a disparate game to physically construct a contraption that
would be able to churn out Newton’s method of divided differences.
And so it proved out to be.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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